Sistemas y Señales Biomédicos

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-04-05

Sistemas y Señales Biomedicos - SYSB

Digital Filters

Introduction

  • The Z-transform is a fundamental tool in digital signal processing (DSP), widely used in biomedical signal processing.
  • It allows analysis of discrete-time biomedical signals such as:
    • ECG (Electrocardiogram): Heart activity
    • EEG (Electroencephalogram): Brain waves
    • EMG (Electromyogram): Muscle activity
  • It helps design digital filters, analyze system stability, and perform signal reconstruction.

Definition of the Z-Transform

The Z-transform of a discrete-time signal \(x[n]\) is defined as:

\[X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}\]

where: - \(X(z)\) is the Z-domain representation of \(x[n]\). - \(z\) is a complex variable: \(z = r e^{j\omega}\), where \(r\) is the magnitude and \(\omega\) is the frequency. - The Z-transform provides a way to study frequency characteristics of biomedical signals.

The Inverse Z-Transform

  • The Inverse Z-Transform allows us to retrieve the original discrete-time signal from \(X(z)\).
  • It is given by:

\[x[n] = \frac{1}{2\pi j} \oint_{C} X(z) z^{n-1} dz\]

  • Common methods to compute the inverse Z-transform:
    1. Partial Fraction Expansion: Used when dealing with rational functions.
    2. Power Series Expansion: Expanding \(X(z)\) as a series to identify coefficients.
    3. Residue Method: Using contour integration for more complex cases.

Example: Step Response in Biomedical Systems

  • A simple low-pass filter used for smoothing an ECG signal has:

\[H(z) = \frac{1}{1 - 0.9z^{-1}}\]

Expanding in a power series:

\[H(z) = 1 + 0.9z^{-1} + 0.81z^{-2} + 0.729z^{-3} + ...\]

The inverse Z-transform reveals an exponentially decaying impulse response, modeling the smoothing effect of the filter.

Relationship with Sampling Method

  • Biomedical signals originate in continuous time and must be sampled for digital processing.
  • Sampling frequency (\(f_s\)) determines the accuracy of the digital representation:

\[T_s = \frac{1}{f_s}\]

  • The Z-transform relates to sampling through the Discrete-Time Fourier Transform (DTFT):

\[X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}\]

where \(X(e^{j\omega})\) is obtained by evaluating the Z-transform along the unit circle: \(z = e^{j\omega}\).

Example: ECG Sampling Rate

  • Standard ECG systems sample at 250 Hz or 500 Hz.
  • The Nyquist frequency is 125 Hz or 250 Hz, respectively.
  • Using the Z-transform, we analyze how filters modify the frequency content of ECG signals.

Region of Convergence (ROC)

  • The Region of Convergence (ROC) determines where the Z-transform sum converges.
  • The ROC provides insights into:
    • System stability
    • Causality
    • Invertibility

Types of ROC:

  1. Right-sided signals (causal systems):
    • The ROC is outside the outermost pole.
    • The system is stable if the ROC includes the unit circle (\(|z| = 1\)).
  2. Left-sided signals (anti-causal systems):
    • The ROC is inside the innermost pole.
  3. Two-sided signals:
    • The ROC lies between poles.

Example: Stability of a Biomedical Filter

  • A high-pass ECG filter with transfer function:

\[H(z) = \frac{1 - z^{-1}}{1 - 0.95 z^{-1}}\]

  • The pole at 0.95 means the system is stable since \(|0.95| < 1\).
  • If the pole were outside the unit circle, the system would be unstable.

Relationship with Convolution

  • In biomedical DSP, filtering operations rely on convolution.
  • Convolution in time domain:

\[y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]\]

  • Multiplication in Z-domain:

\[Y(z) = X(z) H(z)\]

  • This simplifies filter design, allowing us to analyze biomedical signals efficiently.

Example: EEG Band-Pass Filtering

  • EEG signals contain different frequency bands:
    • Delta (0.5–4 Hz): Deep sleep
    • Theta (4–8 Hz): Relaxation
    • Alpha (8–12 Hz): Resting state
    • Beta (12–30 Hz): Active thinking
  • A band-pass filter for extracting alpha waves (8–12 Hz) is designed as:

\[H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}\]

The Z-transform allows us to analyze and optimize this filter.

Conclusion

  • The Z-transform is crucial for analyzing and processing biomedical signals.
  • It enables:
    • Stability analysis (Region of Convergence)
    • Filtering and feature extraction (EEG, ECG, EMG signals)
    • Efficient signal convolution
  • The inverse Z-transform reconstructs signals for further analysis.
  • Understanding the Z-transform helps in filter design, denoising, and feature extraction in biomedical applications.

References

  • Oppenheim, A. V., & Schafer, R. W. (2010). Discrete-Time Signal Processing.
  • Ingle, V. K., & Proakis, J. G. (2011). Digital Signal Processing using MATLAB.
  • Rangayyan, R. M. (2015). Biomedical Signal Analysis.